Hi I'm tying to understand notations in Integration, and would really appreciate some help making sure that my understanding is right.(adsbygoogle = window.adsbygoogle || []).push({});

My books writes

Let u and v be functions of x whose domains are an open interval I, and suppose du and dv exist for every x in I.

Then it defines

1) ∫(du)= u + C

2) ∫(c*du) = c*∫(du)

and

3) ∫(cos(u)du = sin u +C

Now i do understand the first 2, but I want to make sure i understand the 3rd rule.

If u is a function of x with the equation u(x)=x^2

Then the derivative

du/dx=2x

The differential

du=u'(x)dx

Now if it's true that du=u'(x)dx

Then it does make sense that ∫du =u+C because ∫du=∫u'(x)*dx and the integral of the derivative if u is u.

But if u=x^2

Then ∫(cos(x^2)*du = ∫(cos(x^2)*(2x)dx and this is = sin(x^2) + C as the statement above says.

because the derivative of sin(x^2) = cos(x^2)*(2x).

Is this the right interpretation?

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# Understanding an integral rule.

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